JOURNAL OF ELECTRON MICROSCOPY TECHNIQUE 7277-282 (1987)

Quasicrystals and Noncrystallographic Symmetry K.H. KUO BeGing Laboratory of Electron Microscopy, Chinese Academy of Sciences, Beijing, People's Republic of China

KEY WORDS Electron microscopy, Electron diffraction

ABSTRACT New findings on quasicrystals with icosahedral, octagonal, decagonal, and dodecagonal symmetries obtained recently in the Beijing Lab- oratory of Electron Microscopy, Chinese Academy of Sciences, are presented. Special emphasis is put on the relation between quasicrystalline and crystal- line structures. The important role played by electron diffraction and high- resolution electron microscopy in revealing these quasiperiodic structures is pointed out.

INTRODUCTION

Quasicrystal, or rather quasiperiodic crys- tal, is fascinating because it is associated with noncrystallographic symmetry. Unlike crystallographic symmetry, which is limited in number, such as the well-known 14 Bra- vais lattices and 32 point groups, and famil- iar to all solid-state scientists, noncrystal- lographic symmetry is, at least in theory, unlimited in number and was quite foreign to most crystallographers before the discov- ery of the first A1-Mn quasicrystal with the icosahedral point group of symmetry but no translation periodicity by Shechtman et al. (1984). Soon after this, quasicrystals with decagonal (Bendersky, 1985) and dodeca- gonal (Ishimasa et al., 1985) rotational sym- metries were found. The latest addition to this group is the octagonal quasicrystal, found recently in this laboratory (Wang et al., 1987). No one will be surprised now if a quasicrystal with seven- or ninefold rota- tional symmetry is reported, and, as a matter of fact, speculation in this direction has al- ready been made (Mackay, 1986).

The Beijing Laboratory of Electron Micros- copy of the Chinese Academy of Sciences was founded in 1985, and since its very start quasicrystals and noncrystallographic sym- metry have been intensively studied, mainly by selected area electron diffraction (SAED). In close colIaboration with the Laboratory of Atomic Imaging of Solids in Shenyang, also a part of the Chinese Academy of Sciences, investigations on the high-resolution elec- tron microscopy (HREM) of these quasicrys- tals have also been carried out. Some recent

findings of these studies will be briefly out- lined below.

Icosahedral quasicrystal The main object of this study (Zhang et al.,

1987) is to ascertain whether the electron diffraction pattern (EDP) displaying fivefold symmetry is a composite EDP of multiple twins (penta-, deca-, or icosa-twins) or a real sign of the icosahedral quasicrystal. Figure l a is the fivefold EDP of the icosahedral phase found in a rapidIy solidified AI-Cr al- loy, whereas Figure lb-e shows the [loll EDPs of the A145Cr7 crystalline phase (a=2.531, b=0.754, c=1.095nm, ,f3=128.71") existing also in this alloy. In the [loll EDP of a single crystal of A145Cr7, there are ten spots (those indexed in Fig. lb) located at more-or-less the same positions as the ten strong spots of the third concentric decagon in the icosahedral phase (Fig. la). Two, three, or four twin variants of A145Cr7 can be iden- tified, respectively, in Figure lc-e, as indi- cated by the arrows in the (010)* direc- tions and also by the (101)" rectangular recip- rocal cells. The ten spots indexed in Figure l b of these twin variants are nearly but not exactly in coincidence and therefore they oc- cur as groups of spots (Fig. le) and have an apparent appearance of fivefold symmetry. But the other spots do not overlap. Figure If is a composite EDP of the icosahedral quasi-

Received July 31, 1987, accepted August 10, 1987. Address reprint requests to Professor K. H. Kuo, Beijing Lab-

oratory of Electron Microscopy, P.O. Box 2724, Beijing, China.

0 1987 ALAN R. LISS, INC.

278 K.H. KUO

Fig. 1. The %-fold EDP of the icosahedral quasicrystal (a) and the [loll EDPs of A145Cr7 (b) and its twins (c-D. The ten spots indexed in b nearly but not exactly coin- cide their counterparts in the twins, which can be noted either by the rectangular cells or by arrows in the re-

spectiye (010): directions. The arrowheads in f mark the 111 and 222 spots of A145Cr7 and they are periodic, while those of the icosahedral quasicrystal are aperiodic. (Courtesy H . Zhang.)

QUASICRYSTALS AND NONCRYSTALLOGRAPHIC SYMMETRY 279

crystal and fivefold twins of A145Cr7 and it is not quite the same as Figure la. Along each of these twofold directions, in addition to the spots occurring with irregular spacings asso- ciated with the golden mean (7) of the icosa- hedral phase, there are spots occurring a4 exactly the 113 and 213 positions of the 333 spot of the A145Cr7 phase (marked with ar- rowheads in Fig.10. This series of EDPs def- initely shows the existence of an icosahedral quasicrystal in the rapidly solidified A145Cr7 alloy, and Figure l a is not a composite EDP of fivefold twins.

In addition to the fivefold twins around the [loll axis parallel to the fivefold axis of the icosahedral phase, threefold twins have also been found in a direction paraIlel to the threefold axis. The icosahedral point group m35 has the order of 120, whereas the mon- oclinic point group 2lm of the Alpj3-7 has the order of 4, so that 30 twin variants should be possible:

2 - m

{- 35) (order 120)

= (2 3) (order 24) 0 (5) (order 5) m 2 2 2 m m m

= {- - -} (order 8) 0 (3) (order 3)

0 15) (order 5)

2 m 0 { 3) (order 3) 0 { 5) (order 5)

= I-} (order 4) 0 (2) (order 2)

It is interesting to note that the transforma- tion of the icosahedral phase into the A145Ci-7 crystalline phase follows the supergroup- subgroup relationship of phase transforma- tion. From this the orientation relationship between them can be derived which has also been proved by experimental observations.

Octagonal quasicrystal In an extensive investigation of the quasi-

crystals in rapidly solidified transition metal- silicon alloys, EDPs exhibiting eightfold symmetry and no translation periodicity have recently been found in Cr-Ni-Si and V- Ni-Si alloys (Wang et al., 19871, as shown in Figure 2a. However, the idea of eightfold symmetry was not new, though such a quasi- crystal has not been found experimentally until now.

Not long after Penrose found his pair of pentagonal tiles, Robert Ammann, quoted re- cently by Penrose (1986), tiled a plane non- periodically with squares and 45" rhombi (see Figure 3a). This was discussed in detail somewhat later by Beenker (1982) from an algebraic point of view. After the discovery of icosahedral and decagonal quasicrystals, many physicists had studied the nonperiodic

Fig. 2. a: Focused-beam EDP of the Cr-Ni-Si octagonal quasicrystal with its eightfold axis parallel to the beam. b: Simulated diffraction pattern of a two-dimensional eightfold quasilat- tice (Fig. 3a). (Courtesy N. Wang.)

280 K.H. KUO

Fig. 3. a: Two-dimensional octagonal quasilattice con- sisting of squares and 45" rhombi. Three generations are drawn with different- thickness lines to show the hierachical nature. b: High-resolution electron micro- scopic image of the octagonal quasicrystal. Square and 45" rhombi are outlined to show the presence of many octagons. The junction of eight rhombi and that of two squares and four rhombi can be found in many places in a. (Courtesy N. Wang.)

tessellation with eightfold rotational sym- metry (Gratias and Cahn, 1986; Janssen, 1986; Steinhardt, 1986; Watanabe et al., 1987).

There are several points of interest to be noted in Figure 3a. First, of course, is the eightfold symmetry. All the line segments lie parallel to an edge of an octagon and on a set of eight straight lines joining at a point where eight 45" rhombi sharing a common vertex. However, only the central one in Figure 3a is the real eightfold axis, though there are many regions of various sizes of local eight- fold symmetry. Secondly, there is no transla.

tional periodicity. However, these squares and 45" rhombi have exact position order- quasiperiodicity-and, as a matter of fact, Figure 3a is drawn by a computer according to a fixed algorithm. Thirdly, this pattern is hierarchic and three generations of squares and rhombi are drawn in Figure 3a in super- position. They are similar and each genera- tion is inflated by a factor of l + & in linear dimension to the next-higher generation. Each square consists of three squares and four rhombi of a lower generation, while each rhombus consists of two squares and three rhombi of a lower generation. This inflation process can continue indefinitely, so does the deflation in the reverse direction. The area ratio of a square to a rhombus is a, which is a natural consequence of eightfold sym- metry (45 "1.

The Fourier transformation of this pattern (Fig. 2b) can easily be calculated, and it cor- responds to the eightfold electron diffraction pattern of the octagonal phase. Levine and Steinhardt (1984) proved that such an aperi- odic pattern can also give sharp diffraction spots. Our calculated data agree fairly well with the experimental results shown in Fig- ure 2a. By high-resolution electron micros- copy, we have also observed lattice images with bright spots forming squares and 45" rhombi, as shown in Figure 3b. The part of the image outlined in Figure 3b consists both of eight 45" rhombi and of four rhombi and two squares sharing a common vertex. Both configurations can be found in the two-di- mensional eightfold quasiperiodic pattern in Figure 3a. All these lead to the conclusion that an eightfold quasicrystal has been ob- served. Analogous to the icosahedral, deca- gonal, and dodecagonal phases, we propose to call it the octagonal phase.

Decagonal quasicrystal The decagonal quasicrystal was known to

exist in a number of Al-transition metal sys- tems, such as Al-Mn (Bendersky, 1985), Al- Fe (Fung et al., 1986), A1-Ru, A1-F't (Bancel and Heiney, 19861, and A1-Pd (Sastry and Suryanarayana, 1986). It should be of inter- est to investigate whether this quasicrystal will exist or not in other Al-transition metal systems. Therefore an extensive study of this has been carried out in this laboratory and new decagonal quasicrystals have been found in A1-Cr (Si), Al-Co (Dong et al., 19871, A1-Ni (Si), A1-Rh, and Al-0s systems. They are summarized in Table 1. In the cases of A1-Cr

QUASICRYSTALS AND NONCRYSTALLOGRAPHIC SYMMETRY 28 1

TABLE I . Transition metals known to form decagonal quasicrystats with aluminum (Below each element, the

crystalline phase in coexistence with the decagonalphase is Liste4

CdSi) Mn Fe Co Ni(Si)

X-Al4Mn' A l , ~ F e 4 ~ A 1 ~ C o 2 ~ Al~(Ni ,S i )2~

Ru Rh Pd A1&Uq A19Rhz4

0 s - Pt A1130S4

'A1,Mn: a = 2.84, b = 1.24 nm. 'A1,3Fe4: CWm: a = 1.549, b = 0.808, c = 1.248 nm, 6 = 107-43'. 3A19C02: P2,/a: a = 0.855, b = 0.629, c = 0.621 nm, 0 = 94.76". 41sostructural with A19C02.

and A1-Ni alloys, some silicon is required to stabilize the decagonal phase. In the A1-Ir alloy, this phase is also expected to exist. Figure 4a is the tenfold EDP of the Al-0s decagonal phase which shows a distinct ten- fold symmetry and is different from the EDP of the icosahedral phase shown in Figure la.

An interesting problem in the two-dimen- sional decagonal phase is the periodicity in the tenfold direction. As already pointed out by Sastry and Suryanarayana (1986), the twofold patterns in the A1-Mn and A1-Pd decagonal phases are different. In these two- fold patterns, the strong spots form a hexa- gon and they correspond to an interplanar spacing of 0.20-0.21 nm. Between two such strong spots in the tenfold direction, there are six spots in the Al-Mn and eight in the A1-Pd decagonal phases. Figure 4b is such a twofold EDP in the Al-Ni(Si) decagonal phase similar to that of the Al-Pd decagonal phase. In other words, the periodicity along the ten- fold axis is different. According to this, the decagonal phases in the Al-transition metal systems can be divided into two groups. The Al-Cr(Si) and A1-Mn decagonal phases be- long to one group with a periodicity of 6x0.21=1.26 nm, and the other decagonal phases to the right of Mn in Table 1 have a periodicity of about 8 x 0.21 = 1.68 nm. Such a difference in periodicity may have some- thing to do with the lattice parameter of the crystalline phase parallel to the tenfold axis of the decagonal phase. In the A1-Mn system, the crystalline phase X-Al4Mn has a c param- eter of 1.24 nm and the periodicity of the decagonal phase is 1.26 nm. On the other hand, the periodicity of the A1-Fe decagonal phase (1.68 nm) is about twice of the b param- eter of A113Fe4 (0.808 nm) . It can be seen from Table 1 that several decagonal phases

coexist with a crystalline phase having the structure of A19C02 and its (100) interplanar spacing is 0.855 x sin 94" = 0.853 nm, again about one-half of the periodicity (1.68 nm) of these decagonal phases. All these seem to indicate that the periodicity of the decagonal phase is closely related to the structure of the crystalIine phase in the respective Al- transition metal systems.

Dodecagonal quasicrystal The first dodecagonal quasicrystal was

found in vapor-deposited Cr-Ni particles (Ishimasa et al., 1985). We have now found such a quasicrystal in rapidly solidified V2Ni and V-Ni-Si alloys (Fig. 5a). As in the Cr-Ni case, the crystalline phase coexisting with the dodecagonal quasicrystal is also the u phase in these alloys. This shows again the close relation between quasicrystalline and crystalline structures. In the u phase, the structural unit is the CN14 polyhedron with

Fig. 4. a: Tenfold EDP of the Al-0s decagonal quasi- crystal. (Courtesy Z.M. Wang.) b: Twofold EDP of the Al- Ni(Si) decagonal quasicrystal. (Courtesy X.Z. Li.)

K.H. KUO 282

Fig. 5. Twelvefold EDP of the V-Ni dodecagonal quas- icrystal. (Courtesy H. Chen.)

two hexagons placed antisymmetrically dis- playing 12-fold symmetry. Moreover, these structural units are arranged in a plane as squares and 60" rhombi which are the main tiles of the nonperiodic pattern with twelve- fold rotational symmetry. Since the u phase occurs in many binary and ternary transi- tion metal alloy systems, more dodecagonal quasicrystals may be discovered in these al- loys after rapid solidification.

CONCLUSIONS

From the above description it is perhaps no exaggeration to maintain that electron mi- croscopy (SAED and HREM) has played an important role in the discovery of quasicrys- tals and noncrystallographic symmetry. Rea- sons to account for this are threefold: First, with the exception of the recently found Tz- Al~LisCu, quasicrystals occur most easily during rapid solidification and the grain size of them seldom exceeds a few microns. Single crystal diffraction studies on these quasicrys- tals can only be carried out by SAED and they gave EDPs with distinct 5, 8-, lo-, and 12-fold rotational symmetries which can be noticed immediately. Moreover, HREM gives directly the quasilattice image in the real space. Secondly, the rapidly solidified struc- ture obtained by direct quenching from the melt is in general inhomogeneous; and amor- phous, quasicrystalline, and crystalline phases very often exist together. However, even in the case where quasicrystal exists only as a minor constituent, it can still be singled out and analyzed by means of SAED

and HREM. Thirdly, a combination of SAED and HREM can provide direct information of the structure of quasicrystal and these two micro techniques are complementary to each other. Therefore, it is to be expected that more new findings on quasicrystals will come to light in the future by means of transmis- sion electron microscopy.

ACKNOWLEDGMENTS

The author would like to thank H. Chen, X.Z. Li, N. Wang, R. Wang, Z.M. Wang, and H. Zhang, who have contributed the mate- rials presented in this review.

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